ON A HARDY TYPE INEQUALITY AND A SINGULAR STURM-LIOUVILLE EQUATION

Abstract

OF THE DISSERTATION On a Hardy type inequality and a singular Sturm-Liouville equation by Hui Wang Dissertation Director: Haim Brezis In this dissertation, we first prove a Hardy type inequality for u ∈ W 0 (Ω), where Ω is a bounded smooth domain in RN and m ≥ 2. For all j ≥ 0, 1 ≤ k ≤ m − 1, such that 1 ≤ j + k ≤ m, it holds that ∂ ju(x) d(x)m−j−k ∈ W k,1 0 (Ω), where d is a smooth positive function which coincides with dist(x, ∂Ω) near ∂Ω, and ∂l denotes any partial differential operator of order l. We also study a singular Sturm-Liouville equation −(x2αu′)′ + u = f on (0, 1), with the boundary condition u(1) = 0. Here α > 0 and f ∈ L2(0, 1). We prescribe appropriate (weighted) homogeneous and non-homogeneous boundary conditions at 0 and prove the existence and uniqueness of H2 loc(0, 1] solutions. We study the regularity at the origin of such solutions. We perform a spectral analysis of the differential operator Lu := −(x2αu′)′ + u under homogeneous boundary conditions. Finally, we are interested in the equation −(|x|2αu′)′ + |u|p−1u = μ on (−1, 1) with boundary condition u(−1) = u(1) = 0. Here α > 0, p ≥ 1 and μ is a bounded Radon measure on the interval (−1, 1). We identify an appropriate concept of solution for this equation, and we establish some existence and uniqueness results. We examine the limiting behavior of three approximation schemes. The isolated singularity at 0 is also investigated.